Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of D4×D8 |
| automorphism | Aut(G) order 32768 |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C22×D4 |
| commutator | C2×C4 | C23×C4 |
| Frattini, cosocle | C22×C4 | C24 |
| prime radical | C22×#41 | C22 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C4×C22⋊C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
489 conjugacy classes of subgroups; only the 171 normal subgroups are shown.
This group has 489 conjugacy classes of subgroups — too many to draw. Only its 171 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 28 points:
(1,2,3,4)(5,6)(7,9)(8,10)(11,12)(14,19)(15,17)(16,23)(20,24)(21,25)(27,28);(5,7,8,11)(6,9,10,12)(13,14,18,21)(15,19,17,25)(16,24,23,27)(20,26,28,22);(6,10)(9,12)(13,15,18,17)(14,19,21,25)(16,22,23,26)(20,27,28,24);(13,16,18,23)(14,20,21,28)(15,22,17,26)(19,27,25,24)Polycyclic presentation:
2^8|a^2=g,b^2=f,c^2=h,d^2=h,e^2=h,b^a=be,c^a=cf,d^a=dh,e^a=eh,d^b=de,e^b=eh,e^d=eh70 conjugacy classes — view the full character table.
| maximal subgroup of | #220574, #220576, #220592, … |
| maximal normal subgroup of | #220574, #220576, #220592, … |
| maximal quotient of | #224949, #224973, #224997, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 15561);